On parallel p-equidistant ruled surfaces by using mofied orthogonal frame with curvature in E³

In this paper, it is investigated Ruled surfaces according to modified orthogonal frame with curvature in 3-dimensional Euclidean space. Firstly, we give apex angle, pitch and drall of closed ruled surface in E³. Then, it characterized the relationship between these invariant of parallel p-equidistant ruled surfaces.

A new geometry definition and generation method for a face gear meshed with a spur pinion

The conventional tooth surface of a face gear is difficult to manufacture, and the cutter for the face gear cutting is not uniform even though the parameters of the pinion mating with the face gear slightly change. Based on the analysis of the geometry features of the tooth surface, a new developable ruled surface is defined as the tooth flank of the face gear, for which the most important geometry feature is that the flank could be represented by a family of straight lines, hence it could be generated by a straight-edged cutter. The mathematical models of the new ruled tooth surface, the cutter and the generation method are presented, the deviation between the ruled surface and the conventional surface, the correction of the ruled surface to reduce the deviation are investigated through numerical examples. The manufacturing process is simulated by VERICUT software, and the results demonstrate that even when the principle deviation is added to the machined deviation, the absolute deviation is on the micro-scale. The meshing and contact simulation shows that the new surface could obtain good meshing performance when the number of face gear teeth is greater than three times the number of pinion teeth. This research provides a new method for manufacturing face gears.

A new characterization of null scrolls as parallel surfaces

In this paper, we present a theorem which explains the necessary onditions for parallel surfaces of null scrolls to be surfaces again. Then, we give some properties of parallel ruled surfaces such as the frame fields in terms of the original base curve’s frame vectors, the cases of cross products of these frames’ vector fields. Also, we show the parallel ruled surfaces to be developable from a different point of view. Additionally, we obtain a relation between Gauss curvatures and mean curvatures. Finally, we characterize the striction curve of the parallel ruled surface. In the example, we draw a graph for a particular null scroll and its parallel surface.

Generalized normal ruled surface of a curve in the Euclidean 3-space

In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.

A Novel Error Compensation Method of Five-axis Flank Milling of Ruled Surface by Modifying Tool Path

Five-axis flank milling is widely used in the aerospace and automotive industry. However, diverse sources of errors prevent the improvement of machining accuracy. This paper proposes a novel error compensation method for five-axis flank milling of ruled surface by modifying the original tool path according to the error distribution model. The method contains three steps: First, the errors at the middle of the straight generatrix on the machined surface are calculated according to error distribution, and the corresponding normal vectors are obtained by geometric calculation. Second, multi-peaks Gaussian fitting method is utilized to make connections between parameters in the original tool path and error distribution. Finally, the new tool path is generated by adjusting original tool path. Machining experiments are performed to test the effectiveness of the proposed error compensation method. The error distribution after compensation shows that the average error decreases 74%, and the maximum error (contains overcutting and undercutting) decreases 26%. Results show that the proposed error compensation method is effective to improve the accuracy for five-axis flank milling.

The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational or Ruled Types

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.